Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation

This paper develops a general framework for the well-posedness, regularity, and strong approximation of the stochastic reaction–diffusion equation (SRDE) with superlinear drift and diffusion coefficients. We first extend the well-posedness results in \emph{W. Liu and M. Röckner, J. Funct. Anal., 2902–2922, 2010} and \emph{W. Liu, J. Differential Equations, 572–592, 2013} to the case of superlinear diffusion in the Gelfand triple $V\hookrightarrow H\hookrightarrow V^{\ast }$, with $V$ equipped with the norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_V$, and derive a moment estimate by establishing a new Itô formula for $\mathrm{\parallel }X{\mathrm{\parallel }}_V^p$ with general $p\ge 2$. We then apply this abstract result to the SRDE, establish higher spatial regularity ${\dot{H}}^{1+\gamma }$ for any $\gamma \in [0,1]$ whenever the initial datum lies in the same Sobolev space, and obtain temporal Hölder regularity. Finally, we construct a family of tamed finite element methods (tamed-FEMs) for the SRDE under general assumptions on the tamed functions, derive their long-time unconditional stability, and establish optimal strong convergence rates. To our knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.